1. Field of the Invention
The present invention relates to an equalizer in a receiver of a digital communications system, and more particularly, to blind equalization of modulated data with a DC offset.
2. Description of the Related Art
In many digital communications systems, a source generates digital information for transmission to multiple destination receivers. A transmitter processes the digital information into an encoded (e.g., error-correction encoded) and/or packetized stream of data. The stream of data is then divided into discrete blocks. Each of the blocks is mapped onto a corresponding one of a sequence of code or symbol values (“symbols”) chosen from a pre-defined alphabet, and generated with a period Ts, sometimes referred to as the “baud” period. Symbols may be modulated by an analog, e.g., radio frequency (RF), carrier, in amplitude, phase, and/or frequency prior to physical transmission through the communication medium. Many methods of mapping exist and are well known in the art, and these pre-defined alphabets are generated based on certain criteria. For example, data may be mapped into symbols of a complex data stream as pairs of in-phase (I) and quadrature phase (Q) component values that are subsequently modulated with an RF carrier.
Various modulation techniques, such as pulse amplitude modulation (PAM), quadrature amplitude modulation (QAM), phase-shift keyed (PSK) modulation, or vestigial sideband (VSB) modulation are known in the art of communications to modulate the carrier. For example, modulation formats such as PAM, QAM, and complex VSB are common formats used for transmission of digital television signals in accordance with, for example, the ATSC standard for digital television, “ATSC Digital Television Standard,” September 1995.
For these modulation techniques, a quadrature oscillator may be employed with a complex RF upconverter to form a modulator. The I-component modulates the cosine component generated by the oscillator and the Q-component modulates the sine component of the oscillator. VSB modulation is a form of single-sideband modulation in which the redundant sideband of a real-valued signal is removed in full by filtering, except for a small vestige of the sideband. For VSB modulation, a complex signal is formed with the Q-component being the Hilbert transform of the I-component (however, the Q-component thus contains no additional user information).
A transmitter may also insert a pilot and/or other reference into the modulated carrier signal prior to transmission to aid the receiver in carrier synchronization and recovery. For example, 10 Megabaud terrestrial broadcast of Digital Television (DTV) signals in the United States employs single-carrier, single sideband modulation (known as 8-VSB for 8-level Vestigial Sideband Modulation). A narrowband pilot tone is inserted into the lower band edge of the 8- or 16-VSB data spectrum, containing about 7.5 percent of the power of the data spectrum, to aid in carrier synchronization. Thus, for VSB modulation, the reference signal is applied as a 1.25 DC offset to the VSB constellation. For 8-VSB with a symbol set (±1, ±3, ±5, ±7) and 16 VSB with a symbol set (±1, ±3, ±5, ±7, ±9, ±11, ±13, ±15) the DC offset applied is 1.25. Other references added may include training sequences to aid in equalization by the receiver.
The modulated carrier signal transmitted through the medium (which may be, e.g., terrestrial, cable, underwater, wire, optical fiber, atmosphere, space, etc.) comprises a series of analog pulses, each analog pulse being amplitude and/or phase modulated by a corresponding symbol in the sequence. The pulse shape used typically extends many symbol periods in time. This introduces the possibility of adjacent pulses corrupting each other, a phenomenon known as inter-symbol interference (ISI). Most propagation mediums introduce signal distortion, and factors that cause distortion include added noise (static), signal strength variations (fading), phase shift variations, and multiple path delays. In addition, front-end circuitry of the receiver and transmitter also introduce distortion and noise to the signal. The presence of distortion, noise, fading and multipath introduced by the overall communication channel (transmitter, receiver and propagation medium) can cause digital systems to degrade or fail completely when the bit error rate exceeds some threshold and overcomes the error tolerance of the system.
A receiver performs several functions to demodulate and decode a received signal. Receiver functions include, for example, tuning and RF demodulation of the received signal to an intermediate frequency (IF) signal, synchronization with the RF carrier, equalization, symbol detection, and decoding.
A complex demodulator translates the received signal from RF to intermediate frequency (IF), and performs complex demodulation of the received signal at IF to near passband employing the locally generated reference for the carrier signal. To perform demodulation, the receiver includes synchronization of the locally generated reference to the carrier of the received signal. As mentioned previously, synchronization may employ a pilot embedded in the received signal to align the locally generated reference with the carrier phase of the received signal.
The demodulated signal is then sampled based on an estimate of the symbol period. Timing recovery estimates the symbol period, and this estimate may be fed back to the complex demodulator and sampler to adjust the sampling rate (e.g., via a sampling phase error).
Equalization of the received samples suppresses the effects of ISI, caused by phenomena such as i) residual timing error, ii) pulse shape/multipath distortion from the propagation channel, and/or iii) approximation of the ideal transmit and receive filters for ease of implementation. Carrier recovery generates estimates for the difference in frequency and phase (collectively referred to as angle θ) of the carrier used to modulate the symbols and the locally generated reference used for demodulation. A slicer examines each sample to generate either a soft or hard decision for the symbol that corresponds to the sample(s) under study. A slicer is a decision device that, responsive to the signal at its input, generates the projection of the nearest symbol value to the input signal from the grid of constellation points. The output of the slicer thus corresponds to one of the allowed, discrete levels. After symbol detection, a decoder reconstructs the transmitted data from the symbol sequence.
Equalization may be accomplished using a filter that has the inverse channel function of the communication channel. An estimate of the transmission characteristics of the communication channel (transfer function or impulse response) is either known or measured, and the equalization filter parameters may be set indirectly based on the estimate. The received signal is then passed through the equalizer, which compensates for the non-ideal communication channel by introducing “distortions” into the received signal which tend to cancel the distortions introduced by the communication channel.
For some digital transmission applications, such as digital television broadcasting, each receiver is in a unique location with respect to the transmitter. Accordingly, the characteristics of the communication channel are not known in advance, and may even change over time. For these applications, the equalizer may typically be an adaptive equalizer having variable filter parameters, or filter tap coefficients (“taps”), that are calculated by the receiver. The prior art includes many different methods for adjusting the equalizer filter parameters to restore signal quality to a performance level acceptable by subsequent error-correction decoding.
In some systems including an adaptive equalizer, the parameters of the equalizer filter(s) are set using a predetermined reference signal transmitted with the data, sometimes referred to as a training sequence. However, many systems may not insert a training sequence, and so a receiver typically employs blind equalization. In blind equalization, the equalizer's filter parameters are derived from the received signal itself, rather than by using a training sequence. In the prior art, it is known to adjust the equalizer parameters blindly based on an error term generated from a given cost criterion. For this blind equalization, either soft or hard decisions, or best estimates, of the original input symbols, are compared with the received signal to derive parameters of the equalizer filter(s).
However, when a signal includes a pilot, or DC offset component, included during modulation for synchronization, the pilot signal component degrades equalization performance of a receiver. Such performance may be significantly degraded when the receiver performs blind equalization. Prior art systems typically notch out the DC component in the frequency domain (i.e., notch filtering), or subtract the DC component from the signal (constellation) in the time domain. In the frequency domain, notch filtering generally introduces distortion and notch noise since the notch filter is not as narrow as the DC component that is filtered out. In the time domain, subtracting the DC component from the signal requires an estimate of the level of DC component at the receiver. The DC component is difficult to estimate since the magnitude of the DC component varies due to channel effects (e.g., noise, dispersion, and gain/attenuation) that vary with time.
A data model for the output samples y(n) of a combined channel-equalizer system is described by equation (1):y(n)=h′s(n)+f′w(n),  (1)where the vector s(n) contains a group of symbols representing the signal of interest coming from a single source or multiple sources, the vector w(n) represents added white noise, the vector f denotes the finite impulse response of the equalizer filter, and the vector h denotes the combined finite impulse response of the channel-receiver system (i.e., the contribution of f with the impulse response of the channel). When f is selected as the optimal solution f*, the combined equalizer-channel response approximates a pure delay and is the inverse of the channel response. The contribution of f* with the impulse response of the channel and receiver is the approximate inverse equalizer solution. The l-th component of the vector s(n) is denoted al(n)+p, where p represents a DC offset component that is included in the signal inserted at the transmitter. The DC offset component seen at the receiver may or may not be equivalent to the DC offset p introduced at the transmitter because of channel effects. Additive perturbations on the channel are included in the noise vector w(n) of dimension N. The noise contribution may be filtered by the receiver filter f of same dimension. The equalizer, through the vector h, processes the source sequence.
One blind cost criterion employed for adaptive equalization is the constant modulus (CM) criterion. The stochastic gradient descent of the CM criterion is known as the Constant Modulus Algorithm (CMA). The CMA algorithm is described in an article by D. N. Godard entitled “Self-Recovering Equalization in Two-Dimensional Data Communication Systems,” IEEE Transactions on Communications, vol 28, no. 11, pp. 1867-1875, October 1980, which is incorporated herein by reference. The CM criterion and CMA algorithm were further developed to de-couple equalization and carrier recovery functions in a receiver. Such use of the CM criterion and CMA algorithm for equalization is described in J. R. Triechler et al., “A New Approach to Multipath Correction of Constant Modulus Signals,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-31, no. 2, April 1983, which is incorporated herein by reference. Systems that use such CMA algorithm for adaptive equalization, are described in U.S. Pat. No. 5,872,815 to Strolle et al.
The CM criterion penalizes the deviation of the dispersion of the magnitude squared of the received signal from a pre-calculated constant referred to as the “dispersion constant” or the “Godard radius.” FIGS. 1A and 1B illustrate that the CM criterion is based on deviation from a “radius” about the origin of, for example, a source constellation. FIG. 1A shows a radius 101 of an 8-PSK (phase-shift keyed) constellation plotted for real (e.g., Re or I) and imaginary (e.g., Im or Q) components. In FIG. 1A, each point (symbol) lies on the circle 104 defined by this radius 101 (termed a constant modulus system), and the CM criterion penalizes distance of a received sample (e.g., sample 102) from this circle 104. Even though the constellation may rotate, the constellation remains on the circle, and so applying a CM criterion to this constellation does not penalize spatial rotation of the constellation due to residual carrier offset. FIG. 1B shows a 16-QAM (quadrature amplitude modulation) constellation plotted for real and imaginary components. In FIG. 1B, groups of points (symbols) lie on corresponding concentric circles 111, 112, and 113. The CM criterion defines a radius 114 of circle 115, which is not necessarily a radius of one of the concentric circles 111, 112 and 113 (termed non-constant modulus), as a “common” radial distance from the origin for the points of the constellation. As with the constellation of FIG. 1A, the CM criterion penalizes distance of a received sample (e.g., sample 103) from this circle 115.
The CM criterion defines a cost function JCM that may be expressed as given in equation (2):JCM=E[(|yn(f)|2−γ)2]  (2)where E[•] denotes the expected value, γ is the dispersion constant (also known as the Godard radius), yn(f) is the discrete value that represents the sampled signal (see, e.g., equation (1), and f represents the linear filter (e.g., equalizer taps) introduced to suppress the ISI. The dispersion constant γ is a quantity that can be determined from the type of modulation employed (e.g., QAM, BPSK, etc.). The dispersion constant γ may be derived by calculation, by experiment, or by a combination of both for a particular implementation. For equation (2), the subscript “n” in the notation, such as yn, indicates that the values are discrete time. Thus, notation of variables such as “y(n)” and “yn” as used herein are equivalent.
If no DC offset is inserted in the symbol sequence at the transmitter, in the absence of noise and with all global impulse responses h reachable, the spike vectors h*=+ek are global minima of the CM cost function with dispersion coefficient γ. The dispersion coefficient γ may be defined given as in equation (3):
                    γ        =                                            E              ⁢                              {                                  a                  4                                }                                                    E              ⁢                              {                                  a                  2                                }                                              .                                    (        3        )            Derivation of global minima of the CM cost function is known in the art and is described in, for example, I. Fijalkow et al., “Adaptive Fractionally Spaced Blind Equalization,” IEEE DSP Workshop, Yosemite, Calif., 1994, which is incorporated herein in its entirety by reference.
Equation (2) may be jointly optimized. Joint optimization allows for optimization of two or more variables of interest together. For example, the discrete value yn(τ,0) may be dependent upon timing τ and phase θ. Substitution of yn(τ,θ) in equation (2) then yields a CM cost function that may be optimized, such as by deriving the gradient, with respect to timing τ and frequency θ.
For a real-valued source, such as VSB, the CM criterion, and its stochastic gradient, may be modified by taking the real part of yn(f) in equation (3). The modified CM criterion is referred to as the single-axis (SA) CM criterion, and is given in equation (4).JSA-CM=E[(Re(yn(f))2−γ)2]  (4)where Re {•} denotes the real-part extraction.
The SA-CM criterion may be defined for VSB signals, such as described in a paper by Shah et al, “Global convergence of a single-axis constant modulus algorithm,” Proceedings of the Statistical Signal and Array Processing Workshop, Pocono Manor, Pa., August 2000, which is incorporated herein by reference. SA-CM for VSB signals and blind cost error terms generated from the stochastic gradient of SA-CM criterion are also described in U.S. Pat. No. 6,418,164, entitled “A REDUCED COMPLEXITY BLIND EQUALIZER FOR MULTI-MODE SIGNALING,” issued on Jul. 9, 2002, which is incorporated herein by reference.
Given a defined cost function, the gradient of the cost function may be derived. The stochastic gradient is an approximation of the true gradient that is calculated by taking the derivative of the cost function without taking the expected value. For example, the stochastic gradient of the CM criterion is known as the CM algorithm (CMA) and is derived by taking the derivative of equation (2) with respect to the variable of interest (for the gradient of the single-axis CM, the derivative is taken of equation (2) with respect to the (equalizer tap) function f).
Unfortunately, a pilot signal affects the process of blind equalization, which for VSB modulation is predominantly due to the effect of the DC offset of the received signal on the CM-derived error term. With an arbitrary DC offset, the CM cost criterion (function) admits local spurious minima in terms of the equalizer function applied to the received signal.